reserve i,j for Nat;

theorem :: Th6:
  for K being Ring,M1,M2 being Matrix of K st len M1=len M2 & width M1=
  width M2 & M1 = M1 + M2 holds M2 = 0.(K,len M1,width M1)
proof
  let K be Ring,M1,M2 be Matrix of K;
  assume that
A1: len M1=len M2 and
A2: width M1=width M2 and
A3: M1 = M1 + M2;
  0.(K,len M1,width M1)=M1+M2+(-M1) by A3,Th2;
  then 0.(K,len M1,width M1)=M2+M1+(-M1) by A1,A2,MATRIX_3:2;
  then 0.(K,len M1,width M1)=M2+(M1+(-M1)) by A1,A2,MATRIX_3:3;
  then
A4: 0.(K,len M1,width M1)=M2+(0.(K,len M1,width M1)) by Th2;
  per cases by NAT_1:3;
  suppose
    len M1 > 0;
    then M2 is Matrix of len M1,width M1,K by A1,A2,MATRIX_0:20;
    hence thesis by A4,MATRIX_3:4;
  end;
  suppose
A5: len M1 = 0;
    then len (0.(K,len M1,width M1)) = 0;
    hence thesis by A1,A5,CARD_2:64;
  end;
end;
